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A phase change is the transition of a substance from one state of matter to another, without changing its chemical composition. For example, when ice melts to water, it undergoes a phase change from solid to liquid. During a phase change, energy is either absorbed or released by the substance. This energy is often referred to as latent heat. Importantly, during a phase change at constant temperature and pressure, the Gibbs Free Energy change (\( \Delta G \)) is equal to zero. This means the change in enthalpy (\( \Delta H \)) is exactly offset by the product of temperature and entropy change (\( T \Delta S \)). Understanding phase changes is crucial for solving many thermodynamic problems related to melting, boiling, and freezing.

Entropy is a measure of the disorder or randomness in a system. The change in entropy (\( \Delta S \)) during a phase change can tell us about the energy distribution before and after the change. When a substance undergoes a phase change from a solid to a liquid, its entropy increases because the molecules become more disordered. Using the relation \( \Delta G = \Delta H - T \Delta S \) and knowing that \( \Delta G = 0 \) during a phase change at constant pressure, we can calculate \( \Delta S \) with the formula: \[ \Delta S = \frac{\Delta H}{T} \] This formula relates entropy change to the enthalpy of fusion and the temperature at which the phase change occurs, allowing us to quantify how disorder increases as a substance melts.

The enthalpy of fusion (\( \Delta H_f \)) is the amount of energy required to change a substance from the solid phase to the liquid phase at constant pressure. It is an important thermodynamic quantity that helps in understanding phase changes. For a given phase change at constant pressure, we use the enthalpy of fusion to determine the energy absorbed per mole of the substance as it melts. This energy is needed to overcome the forces holding the solid structure, allowing the molecules to move more freely and form a liquid. In our exercise, the enthalpy of fusion for gold (Au) is given as \( 12.61 \text{ kJ/mol} \), providing the energy needed to convert solid gold into liquid without changing temperature.

Temperature conversion is often required in thermodynamics to use consistent units, typically Kelvin, in calculations. This is because Kelvin is the SI unit of temperature and ties directly into equations involving energy and entropy. In our scenario, the phase change occurs at \( 1064^{\circ} \text{C} \). To convert Celsius to Kelvin, we use the formula: \[ T(\text{K}) = T(^{\circ}\text{C}) + 273.15 \] Applying this, \( T = 1064 + 273.15 = 1337.15 \text{ K} \). Using Kelvin allows all components in thermodynamic equations to be compatible and ensures accurate calculations of properties like entropy change.